It might be the father's name, obviously. (Although it seems slightly odd that she pinched his surname, but didn't name him as her "husband".)
It might be a related family name, or even one picked out of thin air to avoid bringing shame on the Burns family! (An odd one to pick at random, though?)
Perhaps Mary Ann's birth name was Connon, although she normally used the name Burns - if she was born before her parents' marriage, for instance? (A quick look at Free BMD makes that one look unlikely, I admit.)
But that's all guesswork, unless we can find some sort of link between the Burns and Connon families - if Mary Ann had been a servant in a Connon household in 1871, for instance, the family story would have started to look very likely!
David - have you located Mary Ann/Sarah on other censuses?
Rambler